Trilinear coordinates

Short description: Coordinate system based on distances from the sidelines of a given triangle

In geometry, the trilinear coordinates $x : y : z$ of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio $x : y$ is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices $A$ and $B$ respectively; the ratio $y : z$ is the ratio of the perpendicular distances from the point to the sidelines opposite vertices $B$ and $C$ respectively; and likewise for $z : x$ and vertices $C$ and $A$ .

In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances ( $a'$ , $b'$ , $c'$ ), or equivalently in ratio form, $ka' : kb' : kc'$ for any positive constant $k$ . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive.

Notation

The ratio notation [math]\displaystyle{ x : y : z }[/math] for trilinear coordinates is often used in preference to the ordered triple notation [math]\displaystyle{ (x, y, z), }[/math] with the latter reserved for triples of directed distances [math]\displaystyle{ (a', b', c') }[/math] relative to a specific triangle. The trilinear coordinates [math]\displaystyle{ x : y : z, }[/math] can be rescaled by any arbitrary value without affecting their ratio. The bracketed, comma-separated triple notation [math]\displaystyle{ (x, y, z) }[/math] can cause confusion because conventionally this represents a different triple than e.g. [math]\displaystyle{ (2x, 2y, 2z), }[/math] but these equivalent ratios [math]\displaystyle{ x : y : z = {}\! }[/math][math]\displaystyle{ 2x : 2y : 2z }[/math] represent the same point.

Examples

The trilinear coordinates of the incenter of a triangle $△ ABC$ are $1 : 1 : 1$ ; that is, the (directed) distances from the incenter to the sidelines $BC, CA, AB$ are proportional to the actual distances denoted by $(r, r, r)$ , where $r$ is the inradius of $△ ABC$ . Given side lengths $a, b, c$ we have:

Name; Symbol		Trilinear coordinates	Description
Vertices	$A$	[math]\displaystyle{ 1 : 0 : 0 }[/math]	Points at the corners of the triangle
	$B$	[math]\displaystyle{ 0 : 1 : 0 }[/math]
	$C$	[math]\displaystyle{ 0 : 0 : 1 }[/math]
Incenter	$I$	[math]\displaystyle{ 1 : 1 : 1 }[/math]	Intersection of the internal angle bisectors; Center of the triangle's inscribed circle
Excenters	$I A$	[math]\displaystyle{ -1 : 1 : 1 }[/math]	Intersections of the angle bisectors (two external, one internal); Centers of the triangle's three escribed circles
	$I B$	[math]\displaystyle{ 1 : -1 : 1 }[/math]
	$I C$	[math]\displaystyle{ 1 : 1 : -1 }[/math]
Centroid	$G$	[math]\displaystyle{ \frac1a : \frac1b : \frac1c }[/math]	Intersection of the medians; Center of mass of a uniform triangular lamina
Circumcenter	$O$	[math]\displaystyle{ \cos A : \cos B : \cos C }[/math]	Intersection of the perpendicular bisectors of the sides; Center of the triangle's circumscribed circle
Orthocenter	$H$	[math]\displaystyle{ \sec A : \sec B : \sec C }[/math]	Intersection of the altitudes
Nine-point center	$N$	[math]\displaystyle{ \begin{align}&\cos(B-C) : \cos(C-A) \\ &\qquad : \cos(A-B)\end{align} }[/math]	Center of the circle passing through the midpoint of each side, the foot of each altitude, and the midpoint between the orthocenter and each vertex
Symmedian point	$K$	[math]\displaystyle{ a : b : c }[/math]	Intersection of the symmedians – the reflection of each median about the corresponding angle bisector

Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates $1 : 1 : 1$ (these being proportional to actual signed areas of the triangles $△ BGC, △ CGA, △ AGB$ , where $G$ = centroid.)

The midpoint of, for example, side $BC$ has trilinear coordinates in actual sideline distances [math]\displaystyle{ (0 , \tfrac{\Delta}{b} , \tfrac{\Delta}{c}) }[/math] for triangle area $Δ$ , which in arbitrarily specified relative distances simplifies to $0 : ca : ab$ . The coordinates in actual sideline distances of the foot of the altitude from $A$ to $BC$ are [math]\displaystyle{ (0, \tfrac{2\Delta}{a}\cos C, \tfrac{2\Delta}{a}\cos B), }[/math] which in purely relative distances simplifies to $0 : cos C : cos B$ .^[1]^{:p. 96}

Formulas

Collinearities and concurrencies

Trilinear coordinates enable many algebraic methods in triangle geometry. For example, three points

[math]\displaystyle{ \begin{align} P &= p:q:r \\ U &= u:v:w \\ X &= x:y:z \\ \end{align} }[/math]

are collinear if and only if the determinant

[math]\displaystyle{ D = \begin{vmatrix} p & q & r \\ u & v & w \\ x & y & z \end{vmatrix} }[/math]

equals zero. Thus if $x : y : z$ is a variable point, the equation of a line through the points $P$ and $U$ is $D = 0$ .^[1]^{:p. 23} From this, every straight line has a linear equation homogeneous in $x, y, z$ . Every equation of the form [math]\displaystyle{ lx+my+nz=0 }[/math] in real coefficients is a real straight line of finite points unless $l : m : n$ is proportional to $a : b : c$ , the side lengths, in which case we have the locus of points at infinity.^[1]^{:p. 40}

The dual of this proposition is that the lines

[math]\displaystyle{ \begin{align} p\alpha + q\beta + r\gamma &= 0 \\ u\alpha + v\beta + w\gamma &= 0 \\ x\alpha + y\beta + z\gamma &= 0 \end{align} }[/math]

concur in a point $(α, β, γ)$ if and only if $D = 0$ .^[1]^{:p. 28}

Also, if the actual directed distances are used when evaluating the determinant of $D$ , then the area of triangle $△ PUX$ is $KD$ , where [math]\displaystyle{ K = \tfrac{-abc}{8\Delta^2} }[/math] (and where $Δ$ is the area of triangle $△ ABC$ , as above) if triangle $△ PUX$ has the same orientation (clockwise or counterclockwise) as $△ ABC$ , and [math]\displaystyle{ K = \tfrac{-abc}{8\Delta^2} }[/math] otherwise.

Parallel lines

Two lines with trilinear equations [math]\displaystyle{ lx+my+nz=0 }[/math] and [math]\displaystyle{ l'x+m'y+n'z=0 }[/math] are parallel if and only if^[1]^{:p. 98,#xi}

[math]\displaystyle{ \begin{vmatrix} l & m & n \\ l' & m' & n' \\ a & b & c \end{vmatrix}=0, }[/math]

where $a, b, c$ are the side lengths.

Angle between two lines

The tangents of the angles between two lines with trilinear equations [math]\displaystyle{ lx+my+nz=0 }[/math] and [math]\displaystyle{ l'x+m'y+n'z=0 }[/math] are given by^[1]^:p.50

[math]\displaystyle{ \pm \frac{(mn'-m'n)\sin A + (nl'-n'l)\sin B + (lm'-l'm)\sin C}{ll' + mm' + nn' - (mn'+m'n)\cos A -(nl'+n'l)\cos B - (lm'+l'm)\cos C}. }[/math]

Perpendicular lines

Thus two lines with trilinear equations [math]\displaystyle{ lx+my+nz=0 }[/math] and [math]\displaystyle{ l'x+m'y+n'z=0 }[/math] are perpendicular if and only if

[math]\displaystyle{ ll'+mm'+nn'-(mn'+m'n)\cos A-(nl'+n'l)\cos B-(lm'+l'm)\cos C=0. }[/math]

Altitude

The equation of the altitude from vertex $A$ to side $BC$ is^[1]^:p.98,#x

[math]\displaystyle{ y\cos B-z\cos C=0. }[/math]

Line in terms of distances from vertices

The equation of a line with variable distances $p, q, r$ from the vertices $A, B, C$ whose opposite sides are $a, b, c$ is^[1]^{:p. 97,#viii}

[math]\displaystyle{ apx+bqy+crz=0. }[/math]

Actual-distance trilinear coordinates

The trilinears with the coordinate values $a', b', c'$ being the actual perpendicular distances to the sides satisfy^[1]^{:p. 11}

[math]\displaystyle{ aa' +bb' + cc' =2\Delta }[/math]

for triangle sides $a, b, c$ and area $Δ$ . This can be seen in the figure at the top of this article, with interior point $P$ partitioning triangle $△ ABC$ into three triangles $△ PBC, △ PCA, △ PAB$ with respective areas [math]\displaystyle{ \tfrac{1}{2}aa' , \tfrac{1}{2}bb', \tfrac{1}{2}cc'. }[/math]

Distance between two points

The distance $d$ between two points with actual-distance trilinears $a i : b i : c i$ is given by^[1]^{:p. 46}

[math]\displaystyle{ d^2\sin ^2 C=(a_1-a_2)^2+(b_1-b_2)^2+2(a_1-a_2)(b_1-b_2)\cos C }[/math]

or in a more symmetric way

[math]\displaystyle{ d^2 = \frac{a b c}{4\Delta^2}\left(a(b_1-b_2)(c_2-c_1)+b(c_1-c_2)(a_2-a_1)+c(a_1-a_2)(b_2-b_1)\right). }[/math]

Distance from a point to a line

The distance $d$ from a point $a' : b' : c'$ , in trilinear coordinates of actual distances, to a straight line [math]\displaystyle{ lx+my+nz=0 }[/math] is^[1]^{:p. 48}

[math]\displaystyle{ d=\frac{la'+mb'+nc'}{\sqrt{l^2+m^2+n^2-2mn\cos A -2nl\cos B -2lm\cos C}}. }[/math]

Quadratic curves

The equation of a conic section in the variable trilinear point $x : y : z$ is^[1]^:p.118

[math]\displaystyle{ rx^2+sy^2+tz^2+2uyz+2vzx+2wxy=0. }[/math]

It has no linear terms and no constant term.

The equation of a circle of radius $r$ having center at actual-distance coordinates $(a', b', c')$ is^[1]^:p.287

[math]\displaystyle{ (x-a')^2\sin 2A+(y-b')^2\sin 2B+(z-c')^2\sin 2C=2r^2\sin A\sin B\sin C. }[/math]

Circumconics

The equation in trilinear coordinates $x, y, z$ of any circumconic of a triangle is^[1]^{:p. 192}

[math]\displaystyle{ lyz+mzx+nxy=0. }[/math]

If the parameters $l, m, n$ respectively equal the side lengths $a, b, c$ (or the sines of the angles opposite them) then the equation gives the circumcircle.^[1]^{:p. 199}

Each distinct circumconic has a center unique to itself. The equation in trilinear coordinates of the circumconic with center $x' : y' : z'$ is^[1]^{:p. 203}

[math]\displaystyle{ yz(x'-y'-z')+zx(y'-z'-x')+xy(z'-x'-y')=0. }[/math]

Inconics

Every conic section inscribed in a triangle has an equation in trilinear coordinates:^[1]^{:p. 208}

[math]\displaystyle{ l^2x^2+m^2y^2+n^2z^2 \pm 2mnyz \pm 2nlzx\pm 2lmxy =0, }[/math]

with exactly one or three of the unspecified signs being negative.

The equation of the incircle can be simplified to^[1]^{:p. 210, p.214}

[math]\displaystyle{ \pm \sqrt{x}\cos \frac{A}{2}\pm \sqrt{y}\cos \frac{B}{2}\pm\sqrt{z}\cos \frac{C}{2}=0, }[/math]

while the equation for, for example, the excircle adjacent to the side segment opposite vertex $A$ can be written as^[1]^{:p. 215}

[math]\displaystyle{ \pm \sqrt{-x}\cos \frac{A}{2}\pm \sqrt{y}\cos \frac{B}{2}\pm\sqrt{z}\cos \frac{C}{2}=0. }[/math]

Cubic curves

Many cubic curves are easily represented using trilinear coordinates. For example, the pivotal self-isoconjugate cubic $Z (U, P)$ , as the locus of a point $X$ such that the $P$ -isoconjugate of $X$ is on the line $UX$ is given by the determinant equation

[math]\displaystyle{ \begin{vmatrix}x&y&z\\ qryz&rpzx&pqxy\\u&v&w\end{vmatrix} = 0. }[/math]

Among named cubics $Z (U, P)$ are the following:

Thomson cubic: Z(X(2),X(1)), where X(2) = centroid, X(1) = incenter

Feuerbach cubic: Z(X(5),X(1)), where X(5) = Feuerbach point

Darboux cubic: Z(X(20),X(1)), where X(20) = De Longchamps point

Neuberg cubic: Z(X(30),X(1)), where X(30) = Euler infinity point.

Conversions

Between trilinear coordinates and distances from sidelines

For any choice of trilinear coordinates $x : y : z$ to locate a point, the actual distances of the point from the sidelines are given by $a' = kx, b' = ky, c' = kz$ where $k$ can be determined by the formula [math]\displaystyle{ k = \tfrac{2\Delta}{ax + by + cz} }[/math] in which $a, b, c$ are the respective sidelengths $BC, CA, AB$ , and $∆$ is the area of $△ ABC$ .

Between barycentric and trilinear coordinates

A point with trilinear coordinates $x : y : z$ has barycentric coordinates $ax : by : cz$ where $a, b, c$ are the sidelengths of the triangle. Conversely, a point with barycentrics $α : β : γ$ has trilinear coordinates [math]\displaystyle{ \tfrac{\alpha}{a} : \tfrac{\beta}{b} : \tfrac{\gamma}{c}. }[/math]

Between Cartesian and trilinear coordinates

Given a reference triangle $△ ABC$ , express the position of the vertex $B$ in terms of an ordered pair of Cartesian coordinates and represent this algebraically as a vector [math]\displaystyle{ \vec B, }[/math] using vertex $C$ as the origin. Similarly define the position vector of vertex $A$ as [math]\displaystyle{ \vec A. }[/math] Then any point $P$ associated with the reference triangle $△ ABC$ can be defined in a Cartesian system as a vector [math]\displaystyle{ \vec P = k_1\vec A + k_2\vec B. }[/math] If this point $P$ has trilinear coordinates $x : y : z$ then the conversion formula from the coefficients $k 1$ and $k 2$ in the Cartesian representation to the trilinear coordinates is, for side lengths $a, b, c$ opposite vertices $A, B, C$ ,

[math]\displaystyle{ x:y:z = \frac{k_1}{a} : \frac{k_2}{b} : \frac{1 - k_1 - k_2}{c}, }[/math]

and the conversion formula from the trilinear coordinates to the coefficients in the Cartesian representation is

[math]\displaystyle{ k_1 = \frac{ax}{ax + by + cz}, \quad k_2 = \frac{by}{ax + by + cz}. }[/math]

More generally, if an arbitrary origin is chosen where the Cartesian coordinates of the vertices are known and represented by the vectors [math]\displaystyle{ \vec A, \vec B, \vec C }[/math] and if the point $P$ has trilinear coordinates $x : y : z$ , then the Cartesian coordinates of [math]\displaystyle{ \vec P }[/math] are the weighted average of the Cartesian coordinates of these vertices using the barycentric coordinates $ax, by, cz$ as the weights. Hence the conversion formula from the trilinear coordinates $x, y, z$ to the vector of Cartesian coordinates [math]\displaystyle{ \vec P }[/math] of the point is given by

[math]\displaystyle{ \vec{P}=\frac{ax}{ax+by+cz}\vec{A}+\frac{by}{ax+by+cz}\vec{B}+\frac{cz}{ax+by+cz}\vec{C}, }[/math]

where the side lengths are

[math]\displaystyle{ \begin{align} & |\vec C - \vec B| = a, \\ & |\vec A - \vec C| = b, \\ & |\vec B - \vec A| = c. \end{align} }[/math]

References

↑ ^1.00 ^1.01 ^1.02 ^1.03 ^1.04 ^1.05 ^1.06 ^1.07 ^1.08 ^1.09 ^1.10 ^1.11 ^1.12 ^1.13 ^1.14 ^1.15 ^1.16 ^1.17 ^1.18 William Allen Whitworth (1866) Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise, link from Cornell University Historical Math Monographs.

External links

Weisstein, Eric W.. "Trilinear Coordinates". http://mathworld.wolfram.com/TrilinearCoordinates.html.
Encyclopedia of Triangle Centers - ETC by Clark Kimberling; has trilinear coordinates (and barycentric) for more than 7000 triangle centers

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[WW-1] 1.00 ^1.01 ^1.02 ^1.03 ^1.04 ^1.05 ^1.06 ^1.07 ^1.08 ^1.09 ^1.10 ^1.11 ^1.12 ^1.13 ^1.14 ^1.15 ^1.16 ^1.17 ^1.18 William Allen Whitworth (1866) Trilinear Coordinates and Other Methods of Analytical Geometry of Two Dimensions: an elementary treatise, link from Cornell University Historical Math Monographs.

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Trilinear coordinates

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Contents

Notation

Examples

Formulas

Collinearities and concurrencies

Parallel lines

Angle between two lines

Perpendicular lines

Altitude

Line in terms of distances from vertices

Actual-distance trilinear coordinates

Distance between two points

Distance from a point to a line

Quadratic curves

Circumconics

Inconics

Cubic curves

Conversions

Between trilinear coordinates and distances from sidelines

Between barycentric and trilinear coordinates

Between Cartesian and trilinear coordinates

See also

References

External links

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Trilinear coordinates

Notation

Examples

Formulas

Collinearities and concurrencies

Parallel lines

Angle between two lines

Perpendicular lines

Altitude

Line in terms of distances from vertices

Actual-distance trilinear coordinates

Distance between two points

Distance from a point to a line

Quadratic curves

Circumconics

Inconics

Cubic curves

Conversions

Between trilinear coordinates and distances from sidelines

Between barycentric and trilinear coordinates

Between Cartesian and trilinear coordinates

See also

References

External links

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